I was wondering what would be the exact definition of atonal music. My idea of a tonal music would be that it starts with a tonic and a corresponding chord, then it may go on to modulate to other keys as the composer wills and at the end, it may or may not return to the home key. So what is atonal when even the tonal music need/does not retain the same tonality throughout the piece? Or in other words how would music of Schoenberg differ from the music of Debussy/Chopin?

It is not easy to answer this question. It just depends on how you take the word "atonal". In a wide sense, atonal is anything that isn't tonal... i.e. anything which doesn't have clear cadences (modulations are cadences in other tonalities, which is still very tonal )Or you can say that "atonal" is only the music which tries desperately to avoid any kind of tonality, i.e. Schönberg's music.For example, calling spectral music atonal is not wrong, but the term's connotation is a bit tricky (people think of Schönberg, who has nothing to do with spectral music)Also, modal music (for example Messiaen's) is not tonal, as it doesn't use the classical, cadencing way of writing music, but it does have fix poles of tension releases and it follows clear scales.